Question: The sum of the digits of a two-digit integer is $12$. The value of the integer is $13$ times the tens digit. What is the integer?
Explanation: Let the tens digit of the two-digit integer be $a$, and the units digit be $b$. We know that $a+b=12$ and $10a+b = 13a$. Since $10a+b=13a$, $3a = b$. Therefore, $4a=12$ and $a=3$, so $b = 3(3) = 9$. Therefore, the integer is $\boxed{39}$.